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23.5.10 Turbulence Models

To describe the effects of turbulent fluctuations of velocities and scalar quantities in a single phase, FLUENT uses various types of closure models, as described in Chapter  12. In comparison to single-phase flows, the number of terms to be modeled in the momentum equations in multiphase flows is large, and this makes the modeling of turbulence in multiphase simulations extremely complex.

FLUENT provides three methods for modeling turbulence in multiphase flows within the context of the $k$- $\epsilon$ models. In addition, FLUENT provides two turbulence options within the context of the Reynolds stress models (RSM).

The $k$- $\epsilon$ turbulence model options are:

figure   

Note that the descriptions of each method below are presented based on the standard $k$- $\epsilon$ model. The multiphase modifications to the RNG and realizable $k$- $\epsilon$ models are similar, and are therefore not presented explicitly.

The RSM turbulence model options are:

For either category, the choice of model depends on the importance of the secondary-phase turbulence in your application.



$k$- $\epsilon$ Turbulence Models


FLUENT provides three turbulence model options in the context of the $k$- $\epsilon$ models: the mixture turbulence model (the default), the dispersed turbulence model, or a per-phase turbulence model.

$k$- $\epsilon$ Mixture Turbulence Model

The mixture turbulence model is the default multiphase turbulence model. It represents the first extension of the single-phase $k$- $\epsilon$ model, and it is applicable when phases separate, for stratified (or nearly stratified) multiphase flows, and when the density ratio between phases is close to 1. In these cases, using mixture properties and mixture velocities is sufficient to capture important features of the turbulent flow.

The $k$ and $\epsilon$ equations describing this model are as follows:


 \frac{\partial}{\partial t} ( \rho_m k ) + \nabla \cdot ( \r... ...{t,m}}{\sigma_k} \nabla k \right) + G_{k,m} - \rho_m \epsilon (23.5-88)

and


 \frac{\partial}{\partial t} ( \rho_m \epsilon ) + \nabla \cd... ...on }{k } (C_{1\epsilon}G_{k,m} - C_{2\epsilon}\rho_m\epsilon ) (23.5-89)

where the mixture density and velocity, $\rho_m$ and $\vec v_m$, are computed from


 \rho_m = \sum_{i=1}^{N} \alpha_i \rho_i (23.5-90)

and


 \vec v_m = \frac{ {\displaystyle \sum_{i=1}^{N}} \alpha_i \rho_i \vec v_i} {{\displaystyle \sum_{i=1}^{N}} \alpha_i \rho_i} (23.5-91)

the turbulent viscosity, $\mu_{t,m}$, is computed from


 \mu_{t,m} = \rho_m C_{\mu} \frac{k^2}{\epsilon} (23.5-92)

and the production of turbulence kinetic energy, $G_{k,m}$, is computed from


 G_{k,m} = \mu_{t,m} ( \nabla \vec v_m +{(\nabla \vec v_m)}^T): \nabla \vec v_m (23.5-93)

The constants in these equations are the same as those described in Section  12.4.1 for the single-phase $k$- $\epsilon$ model.

$k$- $\epsilon$ Dispersed Turbulence Model

The dispersed turbulence model is the appropriate model when the concentrations of the secondary phases are dilute. In this case, interparticle collisions are negligible and the dominant process in the random motion of the secondary phases is the influence of the primary-phase turbulence. Fluctuating quantities of the secondary phases can therefore be given in terms of the mean characteristics of the primary phase and the ratio of the particle relaxation time and eddy-particle interaction time.

The model is applicable when there is clearly one primary continuous phase and the rest are dispersed dilute secondary phases.

Assumptions

The dispersed method for modeling turbulence in FLUENT assumes the following:

Turbulence in the Continuous Phase

The eddy viscosity model is used to calculate averaged fluctuating quantities. The Reynolds stress tensor for continuous phase $q$ takes the following form:


 \overline{\overline{\tau}}_q'' = -\frac{2}{3}(\rho_q k_q + \... ...I}} + \rho_q \mu_{t,q} (\nabla \vec U_q + {\nabla \vec U_q}^T) (23.5-94)

where $\vec U_q$ is the phase-weighted velocity.

The turbulent viscosity $\mu_{t,q}$ is written in terms of the turbulent kinetic energy of phase $q$:


 \mu_{t,q} = \rho_q C_{\mu} \frac{k^2_q}{\epsilon_q} (23.5-95)

and a characteristic time of the energetic turbulent eddies is defined as


 \tau_{t,q} = \frac{3}{2} C_{\mu} \frac{k_q}{\epsilon_q} (23.5-96)

where $\epsilon_q$ is the dissipation rate and $C_\mu=0.09$.

The length scale of the turbulent eddies is


 L_{t,q} = \sqrt{\frac{3}{2}}C_{\mu}\frac{k_q^{3/2}}{\epsilon_q} (23.5-97)

Turbulent predictions are obtained from the modified $k$- $\epsilon$ model:


 \frac{\partial}{\partial t} ( \alpha_q \rho_q k_q ) + \nabla... ...{k,q} - \alpha_q \rho_q \epsilon_q + \alpha_q \rho_q \Pi_{k_q} (23.5-98)

and


 \frac{\partial}{\partial t} ( \alpha_q \rho_q \epsilon_q ) +... ...2\epsilon}\rho_q\epsilon_q) + \alpha_q \rho_q \Pi_{\epsilon_q} (23.5-99)

Here $\Pi_{k_q}$ and $\Pi_{\epsilon_q}$ represent the influence of the dispersed phases on the continuous phase $q$, and $G_{k,q}$ is the production of turbulent kinetic energy, as defined in Section  12.4.4. All other terms have the same meaning as in the single-phase $k$- $\epsilon$ model.

The term $\Pi_{k_q}$ can be derived from the instantaneous equation of the continuous phase and takes the following form, where $M$ represents the number of secondary phases:


 \Pi_{k_q} = \sum_{p=1}^{M} \frac{K_{pq}} {\alpha_q \rho_q }... ...vec v_{p}''> + (\vec U_{p}- \vec U_{q}) \cdot \vec v_{\rm dr}) (23.5-100)

which can be simplified to


 \Pi_{k_q} = \sum_{p=1}^{M} \frac{K_{pq}} {\alpha_q \rho_q } ( k_{pq} -2k_q + \vec v_{pq} \cdot \vec v_{\rm dr} ) (23.5-101)

where $k_{lq}$ is the covariance of the velocities of the continuous phase $q$ and the dispersed phase $l$ (calculated from Equation  23.5-109 below), $\vec v_{pq}$ is the relative velocity, and $\vec v_{\rm dr}$ is the drift velocity (defined by Equation  23.5-114 below).

$\Pi_{\epsilon_q}$ is modeled according to Elgobashi et al. [ 97]:


 \Pi_{\epsilon_q} = C_{3\epsilon} \frac{\epsilon_q}{k_q} \Pi_{k_q} (23.5-102)

where $C_{3\epsilon}=1.2$.

Turbulence in the Dispersed Phase

Time and length scales that characterize the motion are used to evaluate dispersion coefficients, correlation functions, and the turbulent kinetic energy of each dispersed phase.

The characteristic particle relaxation time connected with inertial effects acting on a dispersed phase $p$ is defined as


 \tau_{F,pq} = \alpha_p\rho_q K^{-1}_{pq} \left( \frac{\rho_p}{\rho_q}+C_V \right) (23.5-103)

The Lagrangian integral time scale calculated along particle trajectories, mainly affected by the crossing-trajectory effect [ 70], is defined as


 \tau_{t,pq}= \frac{\tau_{t,q}}{\sqrt{{(1+C_{\beta}\xi^2)}}} (23.5-104)

where


 \xi = \frac{ \vert\vec v_{pq}\vert \tau_{t,q}}{L_{t,q}} (23.5-105)

and


 C_{\beta} = 1.8 - 1.35 \cos^2\theta (23.5-106)

where $\theta$ is the angle between the mean particle velocity and the mean relative velocity. The ratio between these two characteristic times is written as


 \eta_{pq} = \frac{\tau_{t,pq}}{\tau_{F,pq}} (23.5-107)

Following Simonin [ 334], FLUENT writes the turbulence quantities for dispersed phase $p$ as follows:


$\displaystyle k_p$ $\textstyle =$ $\displaystyle k_q \left( \frac{b^2+\eta_{pq}}{1+\eta_{pq}} \right)$ (23.5-108)
$\displaystyle k_{pq}$ $\textstyle =$ $\displaystyle 2k_q \left( \frac{b +\eta_{pq}}{1+\eta_{pq}} \right)$ (23.5-109)
$\displaystyle D_{t,pq}$ $\textstyle =$ $\displaystyle \frac{1}{3}k_{pq}\tau_{t,pq}$ (23.5-110)
$\displaystyle D_p$ $\textstyle =$ $\displaystyle D_{t,pq} + \left(\frac{2}{3}k_p - b\frac{1}{3}k_{pq} \right)\tau_{F,pq}$ (23.5-111)
$\displaystyle b$ $\textstyle =$ $\displaystyle (1+C_V){\left(\frac{\rho_p}{\rho_q}+C_V \right)}^{-1}$ (23.5-112)

and $C_V=0.5$ is the added-mass coefficient.

Interphase Turbulent Momentum Transfer

The turbulent drag term for multiphase flows ( $K_{pq}(\vec v_p - \vec v_q)$ in Equation  23.5-7) is modeled as follows, for dispersed phase $p$ and continuous phase $q$:


 K_{pq}(\vec v_p - \vec v_q ) = K_{pq}(\vec U_p - \vec U_q ) - K_{pq} \vec v_{\rm dr} (23.5-113)

The second term on the right-hand side of Equation  23.5-113 contains the drift velocity:


 \vec v_{\rm dr} = -\left(\frac{D_p}{\sigma_{pq}\alpha_p}\na... ...pha_p - \frac{D_q}{\sigma_{pq}\alpha_q}\nabla \alpha_q \right) (23.5-114)

Here $D_p$ and $D_q$ are diffusivities, and $\sigma_{pq}$ is a dispersion Prandtl number. When using Tchen theory in multiphase flows, FLUENT assumes $D_p = D_q = D_{t,pq}$ and the default value for $\sigma_{pq}$ is 0.75.

The drift velocity results from turbulent fluctuations in the volume fraction. When multiplied by the exchange coefficient $K_{pq}$, it serves as a correction to the momentum exchange term for turbulent flows. This correction is not included, by default, but you can enable it during the problem setup.

You can enable the effect of drift velocity by performing the following:

1.   If it is not already done, set the k-epsilon Multiphase Model to Dispersed in the Viscous panel.

2.   Enter the multiphase-options text command in the console window.

define $\rightarrow$ models $\rightarrow$ viscous $\rightarrow$ multiphase-turbulence $\rightarrow$ multiphase-options

/define/models/viscous/multiphase-turbulence> multiphase-options

Enable dispersion force in momentum? [no] yes

Enable interphase k-epsilon source? [no] yes

The effect of the drift velocity is influenced both by the momentum equation and, to a lesser extent, the turbulence equation. Therefore, you should answer yes to both questions to take into account the effect of drift velocity.

$k$- $\epsilon$ Turbulence Model for Each Phase

The most general multiphase turbulence model solves a set of $k$ and $\epsilon$ transport equations for each phase. This turbulence model is the appropriate choice when the turbulence transfer among the phases plays a dominant role.

Note that, since FLUENT is solving two additional transport equations for each secondary phase, the per-phase turbulence model is more computationally intensive than the dispersed turbulence model.

Transport Equations

The Reynolds stress tensor and turbulent viscosity are computed using Equations  23.5-94 and 23.5-95. Turbulence predictions are obtained from


 \frac{\partial}{\partial t} ( \alpha_q \rho_q k_q ) + \nabla... ... k_q ) + (\alpha_q G_{k,q} - \alpha_q \rho_q \epsilon_q) \; +


 \sum_{l=1}^{N} K_{lq} (C_{lq} k_l - C_{ql} k_q) - \sum_{l=1}... ..._q) \cdot \frac{\mu_{t,q}} {\alpha_q \sigma_q} \nabla \alpha_q (23.5-115)

and


 \frac{\partial}{\partial t} ( \alpha_q \rho_q \epsilon_q ) +... ...q G_{k,q} - C_{2\epsilon}\alpha_q \rho_q\epsilon_q \;+ \right.


 \left. C_{3\epsilon}\left(\sum_{l=1}^{N} K_{lq} (C_{lq} k_l ... ...\mu_{t,q}} {\alpha_q \sigma_q} \nabla \alpha_q \right) \right] (23.5-116)

The terms $C_{lq}$ and $C_{ql}$ can be approximated as


 C_{lq} = 2, \;\; C_{ql} = 2 \left(\frac{\eta_{lq}}{1+\eta_{lq}} \right) (23.5-117)

where $\eta_{lq}$ is defined by Equation  23.5-107.

Interphase Turbulent Momentum Transfer

The turbulent drag term ( $K_{pq}(\vec v_p - \vec v_q)$ in Equation  23.5-7) is modeled as follows, where $l$ is the dispersed phase (replacing $p$ in Equation  23.5-7) and $q$ is the continuous phase:


 \sum_{l=1}^{N} K_{lq}(\vec v_l - \vec v_q ) = \sum_{l=1}^{N}... ... U_l - \vec U_q ) - \sum_{l=1}^{N} K_{lq} \vec v_{{\rm dr},lq} (23.5-118)

Here $\vec U_l$ and $\vec U_q$ are phase-weighted velocities, and $\vec v_{{\rm dr},lq}$ is the drift velocity for phase $l$ (computed using Equation  23.5-114, substituting $l$ for $p$). Note that FLUENT will compute the diffusivities $D_l$ and $D_q$ directly from the transport equations, rather than using Tchen theory (as it does for the dispersed turbulence model).

As noted above, the drift velocity results from turbulent fluctuations in the volume fraction. When multiplied by the exchange coefficient $K_{lq}$, it serves as a correction to the momentum exchange term for turbulent flows. This correction is not included, by default, but you can enable it during the problem setup.

The turbulence model for each phase in FLUENT accounts for the effect of the turbulence field of one phase on the other(s). If you want to modify or enhance the interaction of the multiple turbulence fields and interphase turbulent momentum transfer, you can supply these terms using user-defined functions.



RSM Turbulence Models


Multiphase turbulence modeling typically involves two equation models that are based on single-phase models and often cannot accurately capture the underlying flow physics. Additional turbulence modeling for multiphase flows is diminished even more when the basic underlying single-phase model cannot capture the complex physics of the flow. In such situations, the logical next step is to combine the Reynolds stress model with the multiphase algorithm in order to handle challenging situations in which both factors, RSM for turbulence and the Eulerian multiphase formulation, are a precondition for accurate predictions [ 66].

The phase-averaged continuity and momentum equations for a continuous phase are:


 \frac{\partial}{\partial t}(\overline{\alpha_{\rm c}}\rho_{\... ...t (\overline{\alpha_{\rm c}}\rho_{\rm c}\tilde{U}_{\rm c}) = 0 (23.5-119)


 \frac{\partial}{\partial t}(\overline{\alpha_{\rm c}}\rho_{r... ...\tilde{p} + \nabla \cdot \tilde{\tau}^{t}_{\rm c} + F_{\rm Dc} (23.5-120)

For simplicity, the laminar stress-strain tensor and other body forces such as gravity have been omitted from Equations  23.5-119- 23.5-120. The tilde denotes phase-averaged variables while an overbar (e.g., $\overline{\alpha_{\rm c}}$) reflects time-averaged values. In general, any variable $\Phi$ can have a phase-average value defined as


 \tilde{\Phi}_{\rm c} = \frac{\overline{\alpha_{\rm c} \Phi_{\rm c}}}{\overline{\alpha_{\rm c}}} (23.5-121)

Considering only two phases for simplicity, the drag force between the continuous and the dispersed phases can be defined as:


 F_{\rm Dc} = K_{\rm dc} \left [ (\tilde{U}_{\rm d} - \tilde{... ...\rm c}u'_{\rm c}}}{\overline{\alpha_{\rm c}}} \right) \right ] (23.5-122)

where $K_{\rm dc}$ is the drag coefficient. Several terms in the Equation  23.5-122 need to be modeled in order to close the phase-averaged momentum equations. Full descriptions of all modeling assumptions can be found in [ 65]. This section only describes the different modeling definition of the turbulent stresses $\tilde{\tau^t}$ that appears in Equation  23.5-120.

The turbulent stress that appears in the momentum equations need to be defined on a per-phase basis and can be calculated as:


 \tilde{{\tau^t}_{\rm k}} = - \overline{\alpha_{\rm k}} \rho_{\rm k} \tilde{R}_{\rm k, ij} (23.5-123)

where the subscript $k$ is replaced by $c$ for the primary (i.e., continuous) phase or by $d$ for any secondary (i.e., dispersed) phases. As is the case for single-phase flows, the current multiphase Reynolds stress model (RSM) also solves the transport equations for Reynolds stresses $R_{\rm ij}$. FLUENT includes two methods for modeling turbulence in multiphase flows within the context of the RSM model: the dispersed turbulence model, and the mixture turbulence model.

RSM Dispersed Turbulence Model

The dispersed turbulence model is used when the concentrations of the secondary phase are dilute and the primary phase turbulence is regarded as the dominant process. Consequently, the transport equations for turbulence quantities are only solved for the primary (continuous) phase, while the predictions of turbulence quantities for dispersed phases are obtained using the Tchen theory. The transport equation for the primary phase Reynolds stresses in the case of the dispersed model are:


$\displaystyle \frac{\partial}{\partial t}(\overline{\alpha}\rho\tilde{R}_{\rm i... ...}{\partial x_{\rm k}}(\overline{\alpha}\rho\tilde{U}_{\rm k}\tilde{R}_{\rm ij})$ $\textstyle =$ $\displaystyle -\overline{\alpha}\rho \left ( \tilde{R}_{\rm ik}\frac{\partial \... ...e{\alpha} \mu \frac{\partial}{\partial x_{\rm k}} (\tilde{R}_{\rm ij}) \right ]$  
  $\textstyle -$ $\displaystyle \frac{\partial}{\partial x_{\rm k}} [ \overline{\alpha} \rho \ove... ...{\rm i}}{\partial x_{\rm j}} + \frac{\partial u'_{\rm j}}{\partial x_{\rm i}})}$  
  $\textstyle -$ $\displaystyle \overline{\alpha} \rho \tilde{\epsilon_{\rm ij}} + \Pi_{\rm R, ij}$ (23.5-124)

The variables in Equation  23.5-124 are per continuous phase $c$ and the subscript is omitted for clarity. The last term of Equation  23.5-124, $\Pi_{\rm R, ij}$, takes into account the interaction between the continuous and the dispersed phase turbulence. A general model for this term can be of the form:


 \Pi_{\rm R, ij} = K_{\rm dc} C_{\rm 1, dc} (R_{\rm dc,ij} - ... ...m c,ij}) + K_{\rm dc} C_{\rm 2, dc} a_{\rm dc, i} b_{\rm dc,j} (23.5-125)

where $C_{\rm 1}$ and $C_{\rm 2}$ are unknown coefficients, $a_{\rm dc, i}$ is the relative velocity, $b_{\rm dc, j}$ represents the drift or the relative velocity, and $R_{\rm dc, ij}$ is the unknown particulate-fluid velocity correlation. To simplify this unknown term, the following assumption has been made:


 \Pi_{\rm R, ij} = \frac{2}{3} \delta_{\rm ij} \Pi_k (23.5-126)

where $\delta_{\rm ij}$ is the Kronecker delta, and $\Pi_k$ represents the modified version of the original Simonin model [ 334].


 \Pi_{\rm kc} = K_{\rm dc} (\tilde{k}_{\rm dc} - 2\tilde{k}_{\rm c} + \tilde{V}_{\rm rel} \cdot \tilde{V}_{\rm drift} ) (23.5-127)

where $\tilde{K}_{\rm c}$ represents the turbulent kinetic energy of the continuous phase, $\tilde{k}_{\rm dc}$ is the continuous-dispersed phase velocity covariance and finally, $\tilde{V}_{\rm rel}$ and $\tilde{V}_{\rm drift}$ stand for the relative and the drift velocities, respectively. In order to achieve full closure, the transport equation for the turbulent kinetic energy dissipation rate ( $\tilde{\epsilon}$) is required. The modeling of $\tilde{\epsilon}$ together with all other unknown terms in Equation  23.5-127 are modeled in the same way as in [ 65].

RSM Mixture Turbulence Model

The main assumption for the mixture model is that all phases share the same turbulence field which consequently means that the term $\Pi_{\rm R}$ in the Reynolds stress transport equations (Equation  23.5-124) is neglected. Apart from that, the equations maintain the same form but with phase properties and phase velocities being replaced with mixture properties and mixture velocities. The mixture density, for example, can be expressed as


 \rho_{\rm m} = \sum_{\rm i=1}^N{\overline{\alpha_{\rm i}}} \rho_i (23.5-128)

while mixture velocities can be expressed as


 \tilde{U}_{\rm m} = \frac{\sum_{\rm i=1}^N{\overline{\alpha_... ...}_{\rm i}}{\sum_{\rm i=1}^N{\overline{\alpha_{\rm i}}} \rho_i} (23.5-129)

where $N$ is the number of species.


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